Cyclic group:Z3

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Definition

The cyclic group of order 3 is defined as the unique group of order 3. Equivalently it can be described as a group with three elements $e=x^0,x,x^2$ where $x^l{x}^m=x^{l+m}$ with the exponent reduced mod 3. It can also be viewed as:

• The quotient group of the group of integers by the subgroup of multiples of 3
• The multiplicative group comprising the three cuberoots of unity (as a subgroup of the multiplicative group of nonzero complex numbers)
• The alternating group on three elements
• The group of orientation-preserving symmetries (rotational symmetries) of the equilateral triangle

Properties

Cyclicity

This particular group is cyclic

Abelianness

This particular group is Abelian

Nilpotence

This particular group is nilpotent

Solvability

This particular group is solvable

Simplicity

This particular group is simple: it has no proper nontrivial normal subgroup